# The Stress and Pressure Page

copyright by Philip A. Candela, 1997.

*under construction*

### Force

A Force is a push or a pull, and an unbalanced force produces an acceleration (i.e., a change in velocity over time). Force is a vector quantity; that is, a force is characterized by both magnitude and direction. Note that, for a body traveling at a constant velocity (i.e. a constant speed and in a constant direction), no unbalanced forces are required. That is, motion alone does not require a force; acceleration demands that unbalanced forces exist.

### Stress

When we discuss the forces that act inside deforming materials, we first define a free body, which represents a small element of the material/body in question (e.g. a crystal, a fold belt, a mass of granite magma, an abitrary volume of river water) that has been "cut out". We replace the internal forces by forces distributed over the surfaces of the "free body". When a force is applied to the surface of a free body, we can ratio the force to the area. This ratio is referred to as a stress, S. If the area is broken into innumerable infinitesimal areas, the stress is given by:

S = lim F/A
A=>0

Like force and velocity, stress is a vector quantity. The stress vector can be resolved into a component perpendicular to the surface (the normal stress), and a component parallel to the surface (the tangential, or shear stress)

The stress field about any point in the body may be isotropic (also called hydrostatic) or anisotropic (non-hydrostatic). If we imagine a simple, infinitesimal cube, with its sides oriented orthogonally to the three cartesian coordinates, we can define a normal stress, S1, S2, S3 perpendicular to each pair of sides of the cube. These stresses are defined so that S1>S2>S3. Note that stress has units of Pascals = N/m², (where N = newton). The Pascal is the SI unit of pressure; a Newton is defined as the force necessary to accelerate a one kilogram mass by one meter per second, per second. Twiss and Moores¹ note that the weight (which of course is a force = mg, where g = acceleration due to gravity) of an apple is about one newton! A pressure of 1 bar (nominally, atmospheric pressure at sea level) is the equivalent of 100,000 apples per square meter.

Generally, we might ask what the state of stress might be for any arbitrarily defined plane that passes through the point in question (say, a bedding plane). To answer this type of question, the concept of a tensor was introduced. We use a second rank tensor to determine the magnitude and orientation of a stress vector acting on any given plane, and the tensor can be represented by a 3 x 3 matrix. a Vector is a first rank tensor, and the three components of a vector can be represented by a column matrix. A tensor of rank zero is a scalar quantity (such as speed, or temperature).

Because the "stress tensor" is discussed in structural geology, geophysics, fluid dynamics etc., I will say something about this sometimes mysterious, but in fact, rather simple concept. When the stress tensor matrix, which mathematically desribes the state of stress around a point, is multiplied (dot product) by the column vector which represents the unit vector that is normal to the surface of interest, the product is another vector which represets the stress vector acting on that surface. Putting it poetically, the state of stress around a point, and hence the matrix itself, holds the "potential" for a stress vector that may act on any surface that may pass through the point in question; however, the stress is only "realized" when one plane (defined by the surface normal) is identifed, by writing the column vector (the three numbers in the column matrix (vector) are the magnitudes of the x,y,z components of the surface normal, written in a column). The general form of a stress tensor is given by:

|s11 s21 s31|
|s12 s22 s32| = S(m,n)
|s13 s23 s33|

of which an example is

|  2   1 -3|
|  1   1   2|
|-3   2   1|

Now, of these, which numbers represent the normal stresses and which represent the shear stresses?
If this stress tensor is dotted against a surface normal, such as:

n = (3/7)i + (6/7)j + (2/7)k

where i,j,k are unit vectors along the x,y, and z axes, then, e.g., the x-component of the stress vector is given by:

S(x1) = (3/7)s11 + (6/7)s21 + (2/7)s31.

and s11 = 2, s21 = 1 and s31 = -3. The other two components of the stress vector are defined similarly, yielding:

S = SmnNn = (6/7)i + (13/7)j + (5/7)k.

The indices (m,n) after S are the subscripts of the elements of the stress tensor, and the i,j,k in this equation are unit vectors.

### Pressure

When the stress field is hydrostatic, S1=S2=S3 = P, where P = pressure. Pressure is a scalar (non-directional) quantity; stress is directional. Only when the stress field is isotropic does a thermodynamic pressure exist. A concept called the mean pressure is defined for non-hydrostatic conditions: P(mean) = 1/3{S1+S2+S3}; however, other than the fact that this quantity has the word "pressure" in its name, I see no real reason to consider the "mean pressure" to be a "thermodynamic pressure" in any sense.

Anytime the stress is nonhydrostatic in a rock system (i.e., most of the time), then, strictly, there is no "pressure". Further, there is no unambiguous "Gibbs Free Energy", since G = U + PV -TS. Helmholtz Free Energy can be defined unambigously under non-hydrstatic conditions, however. The stress that defines the thermodynamic properties of solids under nonhydrostatic conditions is the microscopic stress, and I agree with Dahlen (AJS, 1992, pp184-198) who states: "In fact, there is no global equilbrium condition that depends only on the macroscopic principle stresses in addition to the fluid pressures, and the temperature, T. Local equilbrium on the scale of individual solid grains is the only equilbrium possible..."Hence, there is a limit to how precisely geological thermodynamics can be related to

### Notes

Note that the stress tensor is symmetrical about a NW-SE diagonal. This must be so if the shear stresses are such that the free body does not spin (i.e., accelerate angularly). Therefore, only six of the nine elements of the stress tensor are independent. The stress ellipse cannot represent mixed comp and tension.

pure shear (s1=-s3; s2=0)

deviatoric stress = matrix, not a number
differential stress = S1 - S3, a positive scalar quantity
effective stress
Mohr circle: center of circle = mean stress; radius of circle (height of circle) = max poss shear stress;

Prof Lori Kennedy at UBC has discussion of Stress/Mohr circles that you might want to look at.

FOOTNOTE 1 Technically, these are tractions, not stresses; when we pair tractions up on either side of a body, so as to reduce both the acceleration of the center of mass and the angular acceleration about the the center of mass to zero, the tractions on opposite sides of the body become equal and opposite. These paired tractions are, technically, the stresses. This is discussed nicely in Twiss and Moore's book,Structural Geology (W.H. Freeman & Co., 1992) section 8.2. I see no point in introducing this technicality in a thermodynamics forum, but add it here for completeness. Some notes, and other topics to be dealt with presently:

principal stresses
scalar invariants of 2D and 3D stress matrix; mean normal stress is an inv.
uniaxial compression
uniaxial tension